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Special unitary group
In mathematics, the special unitary group of degree , denoted , is the of with 1. (More may have complex determinants with absolute value 1, rather than real 1 in the special case.) The group operation is . The special unitary group is a of the , consisting of all unitary matrices. As a , is the group that preserves the on \mathbb{C}^n . and hence in terms of preservation of the standard inner product on \mathbb{C}^n , see .}} It is itself a subgroup of the , \operatorname{SU}(n) \subset \operatorname{U}(n) \subset \operatorname{GL}(n, \mathbb{C} ) . The groups find wide application in the of , especially }} in the and }} in . The simplest case, , is the , having only a single element. The group is to the group of s of 1, and is thus to the . Since s can be used to represent rotations in 3-dimensional space (up to sign), there is a from to the }} whose is }. , see .}} is also identical to one of the symmetry groups of s, (3), that enables a spinor presentation of rotations. Properties The special unitary group is a real (though not a ). Its dimension as a is }}. Topologically, it is and . Algebraically, it is a (meaning its is simple; see below). The of is isomorphic to the \mathbb{Z}/n , and is composed of the diagonal matrices for an th root of unity and the n''×''n identity matrix. Its , for }}, is \mathbb{Z}_2 , while the outer automorphism group of is the . A maximal torus, of rank , is given by the set of diagonal matrices with determinant 1. The is the , which is represented by (the signs being necessary to ensure the determinant is 1). The of , denoted by \mathfrak{su}(n) , can be identified with the set of complex matrices, with the regular as Lie bracket. often use a different, equivalent representation: The set of traceless complex matrices with Lie bracket given by times the commutator. Lie algebra The Lie algebra \mathfrak{su}(n) of \operatorname{SU}(n) consists of n \times n matrices with trace zero. This (real) Lie algebra has dimension n^2 - 1 . More information about the structure of this Lie algebra can be found below in the section "Lie algebra structure." Fundamental representation In the physics literature, it is common to identify the Lie algebra with the space of trace-zero Hermitian (rather than the skew-Hermitian) matrices. That is to say, the physicists' Lie algebra differs by a factor of i from the mathematicians'. With this convention, one can then choose generators that are complex matrices, where: : T_a T_b = \frac{1}{2n}\delta_{ab}I_n + \frac{1}{2}\sum_{c=1}^{n^2 -1}\left(if_{abc} + d_{abc}\right) T_c where the are the and are antisymmetric in all indices, while the -coefficients are symmetric in all indices. As a consequence, the anticommutator and commutator are: : \begin{align} \left\{T_a, T_b\right\} &= \frac{1}{n}\delta_{ab} I_n + \sum_{c=1}^{n^2 -1}{d_{abc} T_c} \\ \leftT_b\right &= i \sum_{c=1}^{n^2 -1} f_{abc} T_c\,. \end{align} The factor of i in the commutation relations arises from the physics convention and is not present when using the mathematicians' convention. We may also take : \sum_{c,e=1}^{n^2 - 1} d_{ace}d_{bce} = \frac{n^2 - 4}{n} \delta_{ab} as a normalization convention. Adjoint representation In the }}-dimensional , the generators are represented by }}× }} matrices, whose elements are defined by the structure constants themselves: : \left(T_a\right)_{jk} = -if_{ajk}. The group SU(2) is the following group, : \operatorname{SU}(2) = \left\{ \begin{pmatrix} \alpha & -\overline{\beta} \\ \beta & \overline{\alpha} \end{pmatrix}: \ \ \alpha,\beta \in \mathbb{C}, |\alpha|^2 + |\beta|^2 = 1 \right\}~, where the overline denotes . with If we consider \alpha,\beta as a pair in \mathbb{C}^2 where \alpha=a+bi and \beta=c+di , then the equation |\alpha|^2 + |\beta|^2 = 1 becomes : a^2 + b^2 + c^2 + d^2 = 1 This is the equation of the . This can also be seen using an embedding: the map : \begin{align} \varphi \colon \mathbb{C}^2 &\to \operatorname{M}(2, \mathbb{C}) \\5pt \varphi(\alpha, \beta) &= \begin{pmatrix} \alpha & -\overline{\beta}\\ \beta & \overline{\alpha}\end{pmatrix}, \end{align} where \operatorname{M}(2,\mathbb{C}) denotes the set of 2 by 2 complex matrices, is an injective real linear map (by considering \mathbb{C}^2 to \mathbb{R}^4 and \operatorname{M}(2,\mathbb{C}) diffeomorphic to \mathbb{R}^8 ). Hence, the of to the (since modulus is 1), denoted , is an embedding of the 3-sphere onto a compact submanifold of \operatorname{M}(2,\mathbb{C}) , namely . Therefore, as a manifold, is diffeomorphic to , which shows that is and that can be endowed with the structure of a compact, connected . with The complex matrix: : \begin{pmatrix} a + bi & c + di \\ -c + di & a - bi \end{pmatrix} \quad (a, b, c, d \in \mathbb{R}) can be mapped to the : : a\,\hat{1} + b\,\hat{i} + c\,\hat{j} + d\,\hat{k} This map is in fact an isomorphism. Additionally, the determinant of the matrix is the norm of the corresponding quaternion. Clearly any matrix in is of this form and, since it has determinant 1, the corresponding quaternion has norm 1. Thus is isomorphic to the . Relation to spatial rotations Every unit quaternion is naturally associated to a spatial rotation in 3 dimensions, and the product of two quaternions is associated to the composition of the associated rotations. Furthermore, every rotation arises from exactly two unit quaternions in this fashion. In short: there is a 2:1 surjective homomorphism from SU(2) to ; consequently SO(3) is isomorphic to the SU(2)/{±I}, the manifold underlying SO(3) is obtained by identifying antipodal points of the 3-sphere , and SU(2) is the of SO(3). Lie algebra The of consists of 2\times 2 matrices with trace zero. Explicitly, this means : \mathfrak{su}(2) = \left\{ \begin{pmatrix} i\ a & -\overline{z} \\ z & -i\ a \end{pmatrix}:\ a \in \mathbb{R}, z \in \mathbb{C} \right\}~. The Lie algebra is then generated by the following matrices, : u_1 = \begin{pmatrix} 0 & i \\ i & 0 \end{pmatrix}, \quad u_2 = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}, \quad u_3 = \begin{pmatrix} i & 0 \\ 0 & -i \end{pmatrix}~, which have the form of the general element specified above. These satisfy the relationships u_2\ u_3 = -u_3\ u_2 = u_1~, u_3\ u_1 = -u_1\ u_3 = u_2~, and u_1 u_2 = -u_2\ u_1 = u_3~. The is therefore specified by : \leftu_1\right = 2\ u_2, \quad \leftu_2\right = 2\ u_3, \quad \leftu_3\right = 2\ u_1~. The above generators are related to the by u_1 = i\ \sigma_1~, \, u_2 = -i\ \sigma_2 and u_3 = +i\ \sigma_3~. This representation is routinely used in to represent the of s such as s. They also serve as s for the description of our 3 spatial dimensions in . The Lie algebra serves to work out the }}. The group SU(3) Topology The group is a simply-connected, compact Lie group. Its topological structure can be understood by noting that SU(3) acts transitively on the unit sphere S^5 in \mathbb{C}^3 = \mathbb{R}^6 . The of an arbitrary point in the sphere is isomorphic to SU(2), which topologically is a 3-sphere. It then follows that SU(3) is a over the base S^5 with fiber S^3 . Since the fibers and the base are simply connected, the simple connectedness of SU(3) then follows by means of a standard topological result (the for fiber bundles). The SU(2)-bundles over S^5 are classified by \pi_4(S^3)=\mathbb{Z}_2 , and as \pi_4(SU(3))=\{0\} rather than \mathbb{Z}_2 , SU(3) cannot be the trivial bundle SU(2)\times S^5\cong S^3\times S^5 , and therefore must be the unique nontrivial (twisted) bundle. Representation theory The representation theory of is well understood. Descriptions of these representations, from the point of view of its complexified Lie algebra \operatorname{sl}(3; \mathbb{C}) , may be found in the articles on or . Lie algebra The generators, , of the Lie algebra \mathfrak{su}(3) of in the defining (particle physics, Hermitian) representation, are : T_a = \frac{\lambda_a}{2}~, where , the , are the analog of the for : : \begin{align} \lambda_1 ={} &\begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}, & \lambda_2 ={} &\begin{pmatrix} 0 & -i & 0 \\ i & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}, & \lambda_3 ={} &\begin{pmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 0 \end{pmatrix}, \\6pt \lambda_4 ={} &\begin{pmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \end{pmatrix}, & \lambda_5 ={} &\begin{pmatrix} 0 & 0 & -i \\ 0 & 0 & 0 \\ i & 0 & 0 \end{pmatrix}, \\6pt \lambda_6 ={} &\begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix}, & \lambda_7 ={} &\begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & -i \\ 0 & i & 0 \end{pmatrix}, & \lambda_8 = \frac{1}{\sqrt{3}} &\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -2 \end{pmatrix}. \end{align} These span all of the , as required. Note that are antisymmetric. They obey the relations : \begin{align} \leftT_b\right &= i \sum_{c=1}^8 f_{abc} T_c, \\ \left\{T_a, T_b\right\} &= \frac{1}{3} \delta_{ab} I_3 + \sum_{c=1}^8 d_{abc} T_c, \end{align} or, equivalently, : \{\lambda_a, \lambda_b\} = \frac{4}{3}\delta_{ab} I_3 + 2\sum_{c=1}^8{d_{abc} \lambda_c} . The are the of the Lie algebra, given by : f_{123} = 1 , : f_{147} = -f_{156} = f_{246} = f_{257} = f_{345} = -f_{367} = \frac{1}{2} , : f_{458} = f_{678} = \frac{\sqrt{3}}{2} , while all other not related to these by permutation are zero. In general, they vanish, unless they contain an odd number of indices from the set {2, 5, 7}. of all s are non-vanishing.}} The symmetric coefficients take the values :: d_{118} = d_{228} = d_{338} = -d_{888} = \frac{1}{\sqrt{3}} :: d_{448} = d_{558} = d_{668} = d_{778} = -\frac{1}{2\sqrt{3}} :: d_{344} = d_{355} = -d_{366} = -d_{377} = \frac{1}{2} ~. They vanish if the number of indices from the set {2, 5, 7} is odd. A generic group element generated by a traceless 3×3 Hermitian matrix , normalized as 2}}, can be expressed as a ''second order matrix polynomial in : : \begin{align} \exp(i\theta H) ={} &\leftI\sin\left(\varphi + \frac{2\pi}{3}\right) \sin\left(\varphi - \frac{2\pi}{3}\right) - \frac{1}{2\sqrt{3}}~H\sin(\varphi) - \frac{1}{4}~H^2\right \frac{\exp\left(\frac{2}{\sqrt{3}}~i\theta\sin(\varphi)\right)} {\cos\left(\varphi + \frac{2\pi}{3}\right) \cos\left(\varphi - \frac{2\pi}{3}\right)} \\6pt & {} + \left\sin\left(\varphi - \frac{2\pi}{3}\right) - \frac{1}{2\sqrt{3}}~H\sin\left(\varphi + \frac{2\pi}{3}\right) - \frac{1}{4}~H^{2}\right \frac{\exp\left(\frac{2}{\sqrt{3}}~i\theta \sin\left(\varphi + \frac{2\pi}{3}\right)\right)} {\cos(\varphi) \cos\left(\varphi - \frac{2\pi}{3}\right)} \\6pt & {} + \left\sin\left(\varphi + \frac{2\pi}{3}\right) - \frac{1}{2\sqrt{3}}~H \sin\left(\varphi - \frac{2\pi}{3}\right) - \frac{1}{4}~H^2\right \frac{\exp\left(\frac{2}{\sqrt{3}}~i\theta \sin\left(\varphi - \frac{2\pi}{3}\right)\right)} {\cos(\varphi)\cos\left(\varphi + \frac{2\pi}{3}\right)} \end{align} where : \varphi \equiv \frac{1}{3}\leftH\right) - \frac{\pi}{2}\right. Lie algebra structure As noted above, the Lie algebra \mathfrak{su}(n) of \operatorname{SU}(n) consists of n\times n matrices with trace zero. The of the Lie algebra \mathfrak{su}(n) is \mathfrak{sl}(n; \mathbb{C}) , the space of all n\times n complex matrices with trace zero. A Cartan subalgebra then consists of the diagonal matrices with trace zero, which we identify with vectors in \mathbb C^n whose entries sum to zero. The then consist of all the }} permutations of . A choice of s is : \begin{align} (&1, -1, 0, \dots, 0, 0), \\ (&0, 1, -1, \dots, 0, 0), \\ &\vdots \\ (&0, 0, 0, \dots, 1, -1). \end{align} So, is of }} and its is given by , a chain of }} nodes: ... . Its is : \begin{pmatrix} 2 & -1 & 0 & \dots & 0 \\ -1 & 2 & -1 & \dots & 0 \\ 0 & -1 & 2 & \dots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \dots & 2 \end{pmatrix}. Its or is the , the of the }}- . Generalized special unitary group For a , the generalized special unitary group over ''F, , is the of all s of 1 of a of rank }} over which leave invariant a , of . This group is often referred to as the special unitary group of signature over . The field can be replaced by a , in which case the vector space is replaced by a . Specifically, fix a of signature in \operatorname{GL}(n, \mathbb{R}) , then all : M \in \operatorname{SU}(p, q, \mathbb{R}) satisfy : \begin{align} M^{*} A M &= A \\ \det M &= 1. \end{align} Often one will see the notation without reference to a ring or field; in this case, the ring or field being referred to is \mathbb C and this gives one of the classical . The standard choice for when is : A = \begin{bmatrix} 0 & 0 & i \\ 0 & I_{n-2} & 0 \\ -i & 0 & 0 \end{bmatrix}. However, there may be better choices for for certain dimensions which exhibit more behaviour under restriction to subrings of \mathbb C . Example An important example of this type of group is the \operatorname{SU}(2, 1; \mathbb{Z}i) which acts (projectively) on complex hyperbolic space of degree two, in the same way that \operatorname{SL}(2,9;\mathbb{Z}) acts (projectively) on real of dimension two. In 2005 Gábor Francsics and computed an explicit fundamental domain for the action of this group on . A further example is \operatorname{SU}(1, 1; \mathbb{C}) , which is isomorphic to \operatorname{SL}(2, \mathbb{R}) . Important subgroups In physics the special unitary group is used to represent symmetries. In theories of it is important to be able to find the subgroups of the special unitary group. Subgroups of that are important in are, for }}, : \operatorname{SU}(n) \supset \operatorname{SU}(p) \times \operatorname{SU}(n - p) \times \operatorname{U}(1), where × denotes the and , known as the , is the multiplicative group of all s with 1. For completeness, there are also the and subgroups, : \begin{align} \operatorname{SU}(n) &\supset \operatorname{SO}(n), \\ \operatorname{SU}(2n) &\supset \operatorname{Sp}(n). \end{align} Since the of is }} and of is 1, a useful check is that the sum of the ranks of the subgroups is less than or equal to the rank of the original group. is a subgroup of various other Lie groups, : \begin{align} \operatorname{SO}(2n) &\supset \operatorname{SU}(n) \\ \operatorname{Sp}(n) &\supset \operatorname{SU}(n) \\ \operatorname{Spin}(4) &= \operatorname{SU}(2) \times \operatorname{SU}(2) \\ \operatorname{E}_6 &\supset \operatorname{SU}(6) \\ \operatorname{E}_7 &\supset \operatorname{SU}(8) \\ \operatorname{G}_2 &\supset \operatorname{SU}(3) \end{align} See , and for E6, E7, and G2. There are also the : }}, }}, is the of \operatorname{Sp}(2n, \mathbb{C}) . It is sometimes denoted . The dimension of the -matrices is .}} and }}. One may finally mention that is the of , a relation that plays an important role in the theory of rotations of 2- s in non-relativistic . The group SU(1,1) SU(1,1) = \left \{ \begin{pmatrix}u & v \\ v^* & u^* \end{pmatrix} \in M(2,\mathbb{C}) : \ u u^* - v v^* \ = \ 1 \right \}, where u^* denotes the of the complex number u''. This group is locally isomorphic to and where the numbers separated by a comma refer to the of the preserved by the group. The expression u u^* - v v^* in the definition of is an which becomes an when ''u and v'' are expanded with their real components. An early appearance of this group was as the "unit sphere" of s, introduced by in 1852. Let : j = \begin{pmatrix}0 & 1\\ 1 & 0 \end{pmatrix} , \quad k = \begin{pmatrix}1 & 0\\ 0 & -1 \end{pmatrix} , \quad i = \begin{pmatrix}0 & 1\\ -1 & 0 \end{pmatrix} . Then j k = \begin{pmatrix}0 & -1 \\ 1 & 0 \end{pmatrix} = -i . Also i is a square root of −1 (negative of the identity matrix), while j2 = k2 = identity matrix. Similar to Hamilton's quaternions, here ''q = w'' + ''x i + y'' j + ''z k is a coquaternion with conjugate q'' * = ''w – x'' i – ''y j – z'' k. The elements i, j, and k have the property so that the quadratic form is q q^* \ = \ w^2 + x^2 - y^2 - z^2. Note that the 2-sheet \{x i + y j + z k : x^2 - y^2 - z^2 = 1 \} corresponds to the s in the algebra so that any point ''p on this hyperbola can be used as a pole of a sinusoidal wave according to . The hyperboloid is stable under , illustrating the isomorphism with . The variability of the pole of a wave, as noted in studies of , might view as an exhibit of the elliptical shape of a wave with . The model used since 1892 has been compared to a 2-sheet hyperboloid model. When an element of is interpreted as a , it leaves the stable, so this group represents the s of the of hyperbolic plane geometry. Indeed, for a point 1 in the , the action of is given by : z,1\begin{pmatrix}u & v \\ v^* & u^* \end{pmatrix} \ = \ + v^*, \ vz +u^* \ = \ \left+ v^*}{vz +u^*}, \ 1 \right since in (uz + v^*, \ vz +u^*) \ \thicksim \ \left(\frac{uz + v^*}{vz +u^*}, \ 1 \right). Writing suv + \overline{suv} \ = \ 2 \Re(suv), complex number arithmetic shows : |uz + v^*|^2 \ = \ S + zz^* \quad \text{ and } \quad |vz +u^*|^2 \ = \ S + 1, where S \ = \ vv^*(z z^* + 1) + 2 \Re(uvz). Therefore, zz^* < 1 \implies |uz + v^*| < |vz + u^*| so that their ratio lies in the open disk. Footnotes References Category:Advanced mathematics